A caravan of Central American migrants marching into Mexico bound for the United States grew to at least 5,000 people Sunday despite threats from President Donald Trump to use the U.S. military to seal the border.

The throng, many from Honduras, streamed across a bridge over the Suchiate River connecting Guatemala with Mexico.

The Los Angeles Times, citing Gerardo Hernandez, head of the local government’s emergency services, reported that 7,233 immigrants had registered over the past three days at a shelter in Ciudad Hidalgo. Hernandez said his agency has been asked to help provide the immigrants with food and shelter.

The Washington Post, citing Rodrigo Aveja, one of the group’s organizers, reported Sunday night that the caravan included at least 5,600 people.

Many of the migrants cited poverty, corruption and gang violence in Honduras for their flight. Mexico had been trying to slowly process asylum requests in small groups, in some cases providing 45-day visitor permits. But thousands of the migrants grew impatient, circumventing the bureaucracy and crossing over on makeshift rafts or just swimming into Mexico undeterred by border authorities.

They were cheered on by crowds on the Mexican side who shouted, "Venganse!" – Come on in!

In a pair of Sunday tweets, Trump said "full efforts" were being made to "stop the onslaught of illegal aliens" from entering the U.S. and called the caravan a "disgrace to the Democrat party."

“People have to apply for asylum in Mexico first, and if they fail to do that, the U.S. will turn them away," Trump said. "The courts are asking the U.S. to do things that are not doable!”

Full efforts are being made to stop

the onslaught of illegal aliens from crossing our Souther Border. People have to apply for asylum in Mexico first, and if they fail to do that, the U.S. will turn them away. The courts are asking the U.S. to do things that are not doable!

The Mexican government has warned caravan participants “of grave risks” they could face from human trafficking networks if they illegally enter Mexico. U.S. Homeland Security Secretary Kirstjen Nielsen said Sunday the United States was closely monitoring the caravan's advance.

"We must remain mindful of the transnational criminal organizations and other criminals that prey on the vulnerabilities of those undertaking the irregular migration journey," she said in a statement.

"We fully support the efforts of Guatemala, Honduras and Mexico, as they seek to address this critical situation and ensure a safer and more secure region," she said.

An estimated 3,000 would-be migrants marched out of San Pedro Sula, Honduras, on Oct. 13. Their goal was to walk through Guatemala and Mexico to the United States. Their numbers have ebbed and flowed, sometimes numbering little more than 1,000.

But Sunday the crowd was large and the mood mostly ecstatic.

Olivin Castellanos, 58, a truck driver and mason from Villanueva, Honduras, said he took a raft into Mexico. He hopes to work in construction in the United States.

“No one will stop us, only God,” he said. “We knocked down the door and we continue walking.”

“We are going to get to the border of the U.S.,” Luis Puerto, 39, of Colon, Honduras, said in English. “I am not going to stop. I don’t care if I die.”

Slideshow by photo services

Trump has been pressuring the three countries to halt the caravan, threatening to cut off aid in the region and deploy troops along the border if the caravan makes it that far. He has made the caravan a central theme of the midterm elections, accusing the Democrats of supporting open borders and encouraging mass migration.

"I'm willing to send the military to defend our southern border if necessary, all caused because of the illegal immigration onslaught brought by the Democrats," he said at a campaign rally last week. "They like it. They also figure everybody coming in is going to vote Democrat, you know. Hey, they're not so stupid when you think about it."